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Functionals involving surface curvature are important across a range ofscientific disciplines, and their extrema are representative of physicallymeaningful objects such as atomic lattices and biomembranes. Inspired inparticular by the relationship of the Willmore energy to lipid bilayers, weconsider a general functional depending on a surface and a symmetriccombination of its principal curvatures, provided the surface is immersed in a3-D space form. We compute the first and second variations of this functional,leading to expressions given entirely in terms of the surface fundamentalforms. We then apply the stability criteria afforded by our calculations to ageneralization of the Willmore functional, proving a result regarding thestability of spheres.

On the variation of curvature functionals in space forms with application to a generalized Willmore energy